88 PROF. W. K. CLIFFORD ON THE 



In the case of the two triads, since they are oddly distant 

 from one another their origins must be oddly distant ; that 

 is, they must be distant either i or 3. If they are distant 

 I, neither, both, or one of the origins may appear in ihe 

 statement ; if they are distant 3, neither, both, or one of the 

 obverses : 6 types. Thus we obtain 12 types of purely six- 

 fold statement. 



II. If a sixfold statement contains one pair of obverses, 

 the remaining four marks cannot all be evenly distant from 

 this pair. For in that case they would constitute a group ; 

 and it is easy to see that the marks evenly distant from agroup, 

 whether proper or improper, do not contain apair of obverses. 

 We have therefore only these four cases to consider : — 



(i) The four marks are all oddly distant from the obverses. 



(2) One is evenly distant and three oddly distant. 



(3) Two are evenly distant and two oddly. 

 {4) Three are evenly distant and one oddly. 



In the first case the four marks form a group. If this 

 is a proper group, the pair of obverses must be either the 

 origin and obverse of the group, or a pair of mediates ; 2 

 types. If the group is improper, the pair must be an origin 

 and an obverse ; i type. In the second case we have an 

 origin, an obverse, and a mediate, to which we must add 3 

 marks taken out of the proximates and ultimates. We 

 may add 3 proximates distant respectively i i 3 or 1 3 3 from 

 the mediates (2 types), — or 2 proximates distant respec- 

 tively I I, I 3, 3 3 from the mediate, and with each of these 

 combinations an ultimate distant either i or 3 (6 types). 

 To interchange proximates with ultimates clearly makes no 

 difference ; so that in reckoning the cases of i proximate 

 and 2 ultimates or 3 ultimates, we should find no new 

 types. In the third case we have an origin, an obverse, 

 and two mediates distant 2 from each other ; and to these 

 we have to add either two proximates or a proximate and 



